Some excellent critiques have been provided in this thread already but I would like to offer a different perspective.
In programming terms, the switching argument (see original link) is incorrect because it does not typecheck. And it does not typecheck because variable A is used out of its scope when writing down the (5/4)*A expected value.
Indeed, variable A is tied to a specific random outcome and so it simply does not make sense to refer to it in an expected value ranging over this same outcome.
Trying to formalize the argument in a proof assistant makes the mistake clear and obvious. However, the ambiguity of the English language makes it possible for such typing errors to sneak in undetected.
In programming terms, the switching argument (see original link) is incorrect because it does not typecheck. And it does not typecheck because variable A is used out of its scope when writing down the (5/4)*A expected value.
Indeed, variable A is tied to a specific random outcome and so it simply does not make sense to refer to it in an expected value ranging over this same outcome.
Trying to formalize the argument in a proof assistant makes the mistake clear and obvious. However, the ambiguity of the English language makes it possible for such typing errors to sneak in undetected.